Understanding the Volume of a Sphere Formula Using High School Geometry

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Understanding the Volume of a Sphere Formula Using High School Geometry

mathematicsonline · ⏱ 4:14 · 1.5M مشاهدة · 14 years ago

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4 over 3 pi cubed gives you the volume of sphere but where does the formula come from? first we'll draw perfect sphere and fill in it's volume next we will divide the sphere into equal sized pyramids with square bases so let's take look at one pyramid and remember that its volume is equal to one over three area of the base times height now let's quickly show that the height of all the pyramids that make up the sphere is actually equal to the radius so our equation becomes one over three area of the base times radius and we will simplify this equation by replacing radius with and area of the base with we will be numbering the bases so we start with base one and our equation is base-1 over 3 so in order to find the volume of the sphere we simply have to calculate the volume of the pyramids and add them together giving us simple formula so lining up the pyramids we start from the beginning and we have volume of the sphere is equal to base-1 over three plus base-2 over three plus base-3 over three and we will continue this process up until the last pyramid now we don't know how many pyramids we have so we just let base-n over three simply represent the last pyramid so having added up the pyramids together we have this complete simple formula to work with so next we will use algebra and begin by factoring out one over three and we will also factor out the radius giving us over three times the sum of all bases so now let's concentrate on the sum of all bases and remember these are the bases of the pyramids that make up the sphere and as you can see the bases actually form the surface area of the sphere so the sum of all bases is equal to the surface area which equal to four pi squared now quick explanation for why the surface area is four pi squared is first we'll look at our sphere and take its largest possible circumference with that circumference will make circle and now the amazing fact is the surface area of the sphere is equal to exactly four times the area of the circle the area of each circle is pi squared combine them together and we get four pi squared so now let's go back to our simple formula and replace the sum of all bases with four pi squared now we combined the r's together to get cubed and now we simply rearrange the equation to get four over three pi cubed so four over three pi cubed gives you the volume of any sized sphere that exists